Problem: Simplify and expand the following expression: $ \dfrac{4a - 7}{5a + 6}+\dfrac{2a}{4a - 4} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5a + 6)(4a - 4)$ Multiply the first term by $\dfrac{4a - 4}{4a - 4}$ $ \begin{align*} \dfrac{4a - 7}{5a + 6} \times \dfrac{4a - 4}{4a - 4} & = \dfrac{(4a - 7)(4a - 4)}{(5a + 6)(4a - 4)} \\ & = \dfrac{16a^2 - 44a + 28}{(5a + 6)(4a - 4)}\end{align*} $ Multiply the second term by $\dfrac{5a + 6}{5a + 6}$ $ \begin{align*} \dfrac{2a}{4a - 4} \times \dfrac{5a + 6}{5a + 6} & = \dfrac{(2a)(5a + 6)}{(4a - 4)(5a + 6)} \\ & = \dfrac{10a^2 + 12a}{(4a - 4)(5a + 6)}\end{align*} $ Now we have: $ = \dfrac{16a^2 - 44a + 28}{(5a + 6)(4a - 4)} + \dfrac{10a^2 + 12a}{(4a - 4)(5a + 6)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{16a^2 - 44a + 28 + 10a^2 + 12a}{(5a + 6)(4a - 4)} $ $ = \dfrac{26a^2 - 32a + 28}{(5a + 6)(4a - 4)}$ Expand the denominator: $ = \dfrac{26a^2 - 32a + 28}{20a^2 + 4a - 24}$ Simplify: $ = \dfrac{13a^2 - 16a + 14}{10a^2 + 2a - 12}$